Average Error: 30.6 → 0.6
Time: 7.3s
Precision: 64
\[\frac{1 - \cos x}{\sin x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0186108748404142064 \lor \neg \left(x \le 0.022419207862011653\right):\\ \;\;\;\;\frac{1}{\frac{\sin x}{\log \left(e^{1 - \cos x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]
\frac{1 - \cos x}{\sin x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0186108748404142064 \lor \neg \left(x \le 0.022419207862011653\right):\\
\;\;\;\;\frac{1}{\frac{\sin x}{\log \left(e^{1 - \cos x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\

\end{array}
double f(double x) {
        double r61192 = 1.0;
        double r61193 = x;
        double r61194 = cos(r61193);
        double r61195 = r61192 - r61194;
        double r61196 = sin(r61193);
        double r61197 = r61195 / r61196;
        return r61197;
}
↓
double f(double x) {
        double r61198 = x;
        double r61199 = -0.018610874840414206;
        bool r61200 = r61198 <= r61199;
        double r61201 = 0.022419207862011653;
        bool r61202 = r61198 <= r61201;
        double r61203 = !r61202;
        bool r61204 = r61200 || r61203;
        double r61205 = 1.0;
        double r61206 = sin(r61198);
        double r61207 = 1.0;
        double r61208 = cos(r61198);
        double r61209 = r61207 - r61208;
        double r61210 = exp(r61209);
        double r61211 = log(r61210);
        double r61212 = r61206 / r61211;
        double r61213 = r61205 / r61212;
        double r61214 = 0.041666666666666664;
        double r61215 = 3.0;
        double r61216 = pow(r61198, r61215);
        double r61217 = r61214 * r61216;
        double r61218 = 0.004166666666666667;
        double r61219 = 5.0;
        double r61220 = pow(r61198, r61219);
        double r61221 = r61218 * r61220;
        double r61222 = 0.5;
        double r61223 = r61222 * r61198;
        double r61224 = r61221 + r61223;
        double r61225 = r61217 + r61224;
        double r61226 = r61204 ? r61213 : r61225;
        return r61226;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original30.6
Target0.0
Herbie0.6
\[\tan \left(\frac{x}{2}\right)\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.018610874840414206 or 0.022419207862011653 < x

    1. Initial program 0.9

      \[\frac{1 - \cos x}{\sin x}\]
    2. Using strategy rm

    3. Applied clear-num0.9

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin x}{1 - \cos x}}}\]
    4. Using strategy rm

    5. Applied add-log-exp1.2

      \[\leadsto \frac{1}{\frac{\sin x}{1 - \color{blue}{\log \left(e^{\cos x}\right)}}}\]

    6. Applied add-log-exp1.2

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\cos x}\right)}}\]

    7. Applied diff-log1.3

      \[\leadsto \frac{1}{\frac{\sin x}{\color{blue}{\log \left(\frac{e^{1}}{e^{\cos x}}\right)}}}\]

    8. Simplified1.1

      \[\leadsto \frac{1}{\frac{\sin x}{\log \color{blue}{\left(e^{1 - \cos x}\right)}}}\]

    if -0.018610874840414206 < x < 0.022419207862011653

    1. Initial program 60.0

      \[\frac{1 - \cos x}{\sin x}\]

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0186108748404142064 \lor \neg \left(x \le 0.022419207862011653\right):\\ \;\;\;\;\frac{1}{\frac{\sin x}{\log \left(e^{1 - \cos x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{24} \cdot {x}^{3} + \left(\frac{1}{240} \cdot {x}^{5} + \frac{1}{2} \cdot x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64
  :herbie-expected 2

  :herbie-target
  (tan (/ x 2))

  (/ (- 1 (cos x)) (sin x)))